MixCIT: A Kernel Based Local-Polynomial Debiased Test for Conditional Independence on Mixed-Type Data

Conditional independence testing (CIT) is fundamental to modern statistical inference in areas related to causal discovery and variable selection.

While marginal independence is relatively well-understood, despite multiple advances, no existing non-parametric CIT provides a unified, efficient, and statistically guaranteed solution across heterogeneous data.

We introduce a graph-based test statistic comparing kernel similarities of the response within composite neighborhoods that use exact matching on discrete components and $k_n$-nearest-neighbor matching on continuous ones.

The raw statistic, related to prior constructions, suffices under fully discrete conditioning.

However, when at least one conditioning variable is continuous, we instead use a local-polynomial debiased variant that cancels the local smoothing bias.

We rigorously establish its asymptotic null distribution across all data-type combinations.

We further prove a dimension-free $n^{-1/4}$ detection threshold under local alternatives, eliminating the phase transition that affects geometric estimators in high dimensions.

Finally, we develop efficient algorithms with near-quadratic complexity and analytic graph-based calibration, bypassing the cubic bottlenecks of global kernel methods.